Calabi-Yau Manifold

Contents

Introduction

Manifold can be in one, two, three, ... dimensional space with the special property that it looks flat (Euclidean) in small scale but becomes non-Euclidean (curved) globally as shown as Figure 01. Such property implies smoothness. However, a manifold is characterized as smooth even if it has some points with singularity, e.g., a shape point. The dimension of complex manifold involves real and imaginary axes. Since complex number always shows up in pair such as (x,iy), complex manifold always has even dimension. Riemannian geometry is a generalization of the Euclidean geometry to include manifolds with curvature. It prescribes mathematical formulas to calculate values of angle, length of curves, surface area, and volume. All the information are contained within the metric tensors gij as demonstrated by a very simple example in Eqs.(2)-(6).

Figure 01 2-Dimensional Manifold [view large image]

There is a special kind of Riemann surface, which possesses the property that a projection from one surface onto another preserves the angles between things even though the distance may be distorted. This angle-preserving characteristic is known as conformal mapping, which can simplify

		Eugenio Calabi (Figure 02) was born Jewish Italian in 1923. He is American mathematician and professor emeritus at the University of Pennsylvania since 1964, specializing in differential geometry, partial differential equations and their applications. In 1953 he raised the Calabi's conjecture, which proposed that certain specific geometric structures are allowed under some topological condition (briefly, topology concerns with the overall shape, while geometry are often related to the exact shape and curvature of some object). Shing-Tung Yau (, Figure 03) was born Chinese in 1949. He has been a professor of mathematics at Harvard since 1987 and is the current department chair. In 1976, he proves the existence of the geometric structure as surmised by the Calabi's conjecture. Such structure is now called Calabi-Yau manifold. Its special properties are indispensable for compactification in Superstring Theory.
Figure 02 Eugenio Calabi	Figure 03 Shing-Tung Yau	Yau's perception of the conjecture is somewhat different but equivalent. It asks if there could be gravity even when the space contains no matter - in analogy to the Schwarzschild's solution in vacuum but more general.

Calabi-Yau Manifold for Dummies

The concept of Calabi-Yau manifold can be best explained by defining the terminology, without which it is very difficult to comprehend.


• Topology - The formal definition involves language in "Set theory". Translating into plain English, it is the branch of

  mathematics that studies the properties of spatial objects from their inherent connectivity while ignoring the detailed form. The examples in Figure 04 shows that the 2-dimensional objects are separated into different class (space) according to the number of holes in them. Objects in each space have the same topology as long as they can be moulded into each other by deformation, twisting and stretching but no discontinuous operation such as cutting. Topology in higher dimension is more complicated than the 2-dimensional example, but the general idea remains the same. The classification of topological equivalent objects depends on the Euler characteristic = V - E + F, where (for 1 or 2 dimensional object) the number of vertices V = # of (n+1) or more edges meeting at a point + # of end points (n is the dimension of the object), the number of edges E = # of intersections of 2 faces or more + # of boundaries, and F is the number of faces. Table 01 shows some of the geometric objects

  Figure 04 Topology [view large image]

  together with the Euler characteristic. The objects with the same Euler characteristic belong to the same topological class even though they could look very different.

  
  

  Table 01 Euler Characteristic for some Geometric Objects

• Chern Class - Chern class is used to characterize the differences between two manifolds. If two manifolds have different Chern classes, they cannot be the same, but the converse is not always true: two manifolds can have the same Chern classes and still be different. The number of Chern classes depends on the number of complex dimensions (1 complex dimension implies 2 real dimensions) :
  

  1. For 1 complex dimensional manifold, there is just one Chern class, the first Chern class, which equals the Euler characteristic. In Figure 05, the complex 1-D sphere has 2 spots with zero net flow implying a first Chern class = Euler characteristic = 2, while the torus has no such spots and has first Chern class = Euler characteristic = 0. 2. The 2 complex dimensional manifold has a first and a second Chern class. The first one assigns an integer coefficients to subspaces (the 1-D manifold within the 2-D manifold), while the second one (the last one) always equals to the Euler characteristic. 3. The n complex dimensional manifold has a first and a second, ... nth Chern classes. The

  Figure 05 1-D Chern Classes [view large image]

  first one assigns an integer coefficients to subspaces (the 1-D manifold within the n-D manifold), while the second one assigns another number to subspaces (the 2-D manifold within the n-D manifold), ..., the last one always equals to the Euler characteristic.

  The first Chern class is vanishing (or zero) if all the tangent vectors on a manifold can be oriented to the same direction. This is impossible on a 2-D sphere, but can be done on the surface of a 2-D torus. Chern class is useful for learning the big picture of the manifold, to check out the number and types of sub-objects the manifold can hold. It is something similar to taxonomy, which classifies animals into species, genus, ..., kingdom. BTW, The Chern in Chern class is referred to Shiing-Shen Chern (, 1911 - 2004), who was a Chinese mathematician and the mentor of S. T. Yau.




Curvature - The curvature K of one dimensional object is determined by the formula : K = |d(dy/dx)/dx| / [1+(dy/dx)2]3/2 ---------- (1) while the square of an element of an arc on the curve is : ds2 = [1+(dy/dx)2] dx2 ---------- (2) The factor in front of dx2 is the 1 dimensional metric tensor g11 for a smooth manifold. An Euclidean example is a straight line described by the formula : y = ax + b ---------- (3) which produces K = 0, and ds2 = (1+a2) dx2. For a horizontal line such as the one shown in Figure 06, a = 0 and ds = dx. A circle provides an 1 dimensional non-Euclidean example, it is expressed by the formula : x2 + y2 = r2 ---------- (4) which yields K = 1/r ---------- (5),

Figure 06 Curvature of 1-D Object [view large image]

and ds2 = r2/(r2 - x2) dx2 ---------- (6). Note that there are two singularities at x = r. However, integration from x=-r to x=r returns a finite arc length of s = r.

Bypassing the mathematical formula as mentioned above, the curvature at a point P on a 1 dimensional curve can be defined as the reciprocal of the radius of the circle that best matches a curve at a given point (see insert in Figure 06). Viewing from top, curvature for a convex curve (like a circle) has negative sign, while the concave curve (like the bottom half of a cut-off circle) has positive sign.

Defining curvature for 2 dimensional object is more complicated because there are infinite number of curves that can pass through any point on the surface. Followings are some of the various definitions, which are related to the understanding of Calabi-Yau manifold :

1. Gaussian Curvature - The Gaussian curvature K is defined by multiplying the largest curvature k₁, and smallest curvature k₂ (i.e., K=k₁k₂) of the family of sectional curves tangential to the tangent plane at a given point as shown in Figure 07 at the

saddle point of the hyperbolic paraboloid surface. According to the Gauss-Bonnet Theorem the sum of the Gaussian curvatures over the whole surface equals to 2 multiplied the Euler characteristic , i.e. : A K dA = 2 ---------- (7) This theorem means that the overall curvature of the surface is fixed. Bending and pulling the surface may alter the curvature at every point, but these changes all cancel each other

Figure 07 Curvature of 2-D Object [view large image]

out. A trivial example is the surface of a sphere with unit radius (see insert in Figure 07). Its Gaussian curvature is 1 everywhere and the total surface area is 4. Thus = 2, which is the same as calculated previously by another method.



2. Ricci Curvature - For higher dimensional manifolds (> 2-D), there are more than one tangent plane going through a point on the surface. The Ricci curvature is the average of the Gaussian curvatures associated with all these planes. Thus, a manifold can be Ricci flat (Ricci curvature = 0) without being flat overall. For the 2-D spherical manifold in Figure 07, the Ricci curvature equals to the Gaussian curvature since there is only one tangent plane. The Ricci curvature tensor R_ik appears in the 4 dimensional General Relativity Field Equation as :
R_ik - g_ikR/2 = 8GT_ik/c⁴ ---------- (8)

where

,

R = gikRik is the scalar curvature, gik is the metric tensor in the

square of the line element ds² = g_ikdxⁱdx^k (similar to the 1-D example in Eq.(6)) to be determined by the field equation (8),

is the Christoffel symbol, and Tik is the energy-momentum tensor, the indices

i, k, ... run from 0, ..., 3. Thus a Ricci flat curvature, i.e., R_ik = 0, is a solution to an empty universe for the field equation. Note that identical upper and lower indices stands for summation in N dimensions, e.g., xⁱy_i = x¹y₁ + x²y₂ + x³y₃ for N = 3.

3. Riemann Curvature - This is the curvature that completely determines the form of a manifold. It has 20 components (in 4 dimensions) splitting half into the Ricci tensor and the other half into the Weyl tensor. The Weyl tensor does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force or gravitational radiation. In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero.


• Riemann Surface and Complex Manifolds -

Riemann surface is manifold in complex number involving a pair of real and imaginary axes. Conformal mapping can be performed between Riemann and flat surfaces. This process may change the length scale but the angles between things are always maintained. Such property makes calculations easier than working on non-complex manifold. Riemann surfaces must be orientable meaning they have two sides like a beach ball, rather than just one side like a Mobius strip. In addition, they can be classified by their genus g (number of holes). The Euler characteristic = 2 - 2g. There are complex manifolds in higher dimensions with similar properties, but the different regions or patches of such manifold have to be properly attached in order to be qualified as complex.

Figure 08 Conformal Mapping [view large image]

• Kahler Manifold -
  1. It is a complex manifold.
  2. It is in between the Hermitian and flat manifolds in the sense that for a small displacement the metric g_ik varies as and 0 respectively, while the Kahler manifold changes as ². Such manifold has already many existing mathematical tools for its further study.
  

3. Parallel transport is a process of moving vectors or spinors along a path on a manifold that keeps the lengths as well as the angles between any two vectors unchanged (see Figure 09a for moving a vector on spherical surface by maintaining a southerly direction and tangential to the surface. if the direction changes at the end of a closed path, then there is curvature over the path). 4. Holonomy Group contains all the elements, which are closed paths formed by parallel transport, e.g., the path 1, 2, 3 in Figure 09.

Figure 09 Parallel Transport Symmetry [view large image]

5. For a Ricci flat manifold, a vector or spinor retains its orientation after moving through a closed path. Such spinor is called "covariantly constant spinor" and is a member of the SU(3) (3 complex or 6 real dimensions) holonomy group.

6. J operation is to rotate the vector by 90^o or multiply by i in case of complex dimension.
7. The Kahler manifold has the special property that it is symmetrical under the process of : J operation + parallel transport or vice versa, i.e., JW₁ = W₂ as shown in Figure 09b. This symmetry (known as internal symmetry) imposes another set of constraints on the mathematical formulation, drastically simplifying calculations.




Calabi's Conjecture - It was surmised that whether a certain kind of complex manifold - a space that is compact (finite in extent) and "Kahler" - could satisfy the general topological conditions of vanishing first Chern class and could also satisfy the geometrical condition of having a Ricci-flat metric (excluding the flat torus). Calabi-Yau Manifold - By expressing the conjecture in terms of nonlinear partial differential equations, S. T. Yau was able to prove the existence of many multi-dimensional shapes (now called Calabi-Yau manifolds) that are Ricci-flat, i.e., satisfying the Einstein equation in 3 complex dimensions and empty space, though the precise expressions for the metric tensor gik associated with such manifolds are still unknown. Figure 10 is a rendering

Figure 10 Calabi-Yau Manifold [view large image]

of the 2 dimensional cross-section from a 6 dimensional Calabi-Yau manifold.

The superstring theory has conformal invariance automatically built into it as the world sheet possesses conformal structure. Subsequent investigation shows that in order for a conformal field theory to survive the quantization (via the Feynman's prescription of summing over all the world sheet surfaces with weighting factors) the Calabi-Yau manifold should not be exactly Ricci-flat, but the deviation is not large enough to throw out its adoption to the superstring theory in its original form. Another new discovery is the mirror symmetry, which links two Calabi-Yau manifolds with opposite Euler characteristic (i.e., + and -) to the same conformal field theory (in a space one dimension lower). Such kind of duality always implies a simpler mathematical formulation in one of the alternatives.

Theory of Superstring, and M Theory

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

  Figure 11 11-D Space

  or computer screen, if one axis represents many dimensions. The d11 dimension is the 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

  Figure 13 Superstring Theories [view large image]

  Standard Model with mass < 10 Tev can be considered as massless. String theory with 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 size and mass 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 corresponding to three bosons -
  Figure 15 Gauge Invariance [view large image]	Figure 16 Gauge 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
  Figure 17 Broken Symmetry	12 in the SU(3)xSU(2)xU(1) symmetry groups. The process is called 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
     Figure 18 11th Dimension [view large image]	Figure 19 World Path Dimensions	this M-theory. It is suggested that the "true home" of the 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
  Figure 20 Moduli of a Torus [view large image]	Figure 21 Vacuum Energy	fundamental constants are not constant 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.

The Connections

	The connection between string theory and Calabi-Yau manifold is mainly through supersymetry (Figure 22), although there are other linkages that make it even more desirable. Followings are a brief description of the steps leading to this unique selection, and its other special properties beneficial to the development of the Superstring theory.
Figure 22 Connection 1 to Calabi-Yau Manifold [view large image]

1. String - The initial theory of the bosonic string corresponds in low energy limit to particles with integer spin, i.e., they are all bosons. This theory is plagued with a lot of problems, parts of which can be eliminated by going to a space of 26 dimensions. The remaining problems of tachyon and missing fermions can be resolved only by adding supersymmetry into the theory.
2. Supersymmetry - Supersymmetry is used to add fermions into the bosonic string theory. It requires a ten dimensional space time to eliminate the anomlies, inconsistencies, techyons, etc. Thus, the 26 dimensional space for the bosons has to be compacted preliminarily to ten extra dimensions by rolling up 16 to tiny circles or set to zero. If the N = 1 version of the super generator is selected, then it seems to incorporate the chiral character of the fermions nicely as well (see Table 02).
3. Compactification - The remaining six extra dimensions have to be reduced further to small size. The task became more difficult as the chiral characteristic does not survive the compactification process rendering the theory inconsistent with the real world. It does not work for a simple circle or torus or other more complicate manifolds such as the K3 surface (a 4 real or 2 complex dimensional manifold).
4. Parity Violation - Parity violation in weak interaction is an established fact since 1956. Only the left-handed leptons participated in the interaction. Thus the left- and right-handed leptons are different (chiral), all the realistic theories have to accommondate to this fact. Subsequent investigations show that only the compact (finite) Kahler manifold of SU(3) (6 real or 3 complex dimensions) holonomy with vanishing first Chern class and zero Ricci curvature can preserve the chiral characteristic after compactification.
5. Calabi-Yau Manifold - As mentioned in the section of "Calabi-Yau Manifold for Dummies", all the above-mentioned

requirements are satisfied by the Calabi-Yau manifold as if it is "made to order" for the occasion. By the way, it also correctly reproduce the three generations for the fermions, and is itself a solution of the 6-D field equation in General Relativity (producing the gravitino). However, it took nine years between the proof of the "Calabi-Yau Conjecture" in 1976 and the introduction of this strange geometry object to the Superstring theory in 1985 for the mathematics and physics to connect. Since then there are also Calabi-Yau off-Broadway show, Calabi-Yau painting (Figure 23), and Calabi-Yau joke in a New Yorker satire by Woody Allen - all wanted to cash in on its fame.

Figure 23 Calabi-Yau Monna Lisa [view large image]

Further efforts to reproduce the Standard Model in 1986 and later reveal another connection between the Superstring theory and Calabi-Yau manifold. A new theory such as the Superstring theory should recover all the features of an effective theory at lower level. This test would be a show stopper if it fails. The investigations showed that it is possible to obtain all the 12 gauge fields with the Calabi-Yau manifold. Work is also in progress to calculate the ferimon mass from the manifold - a feat not included in

Figure 24 Connection 2 to Calabi-Yau Manifold [view large image]

the Standard model. Figure 24 shows the new connection, which is also explained in more detail below :

1. Calabi-Yau Manifold (with holes) - It has been shown that there are specific requirements in the Superstring theory leading to the selection of the Calabi-Yau manifold for compactification. It was then discovered that only the manifold with three holds could produce the three generations of fermions as observed in the real world. The presence of holes in the manifold inevitably affects the geometry and topology, which in turn affects physics in such a lucky way as to enable the recovery of the Standard model from the Superstring theory.
2. Gauge Fields - It is found that the SU(3)xSU(2)xU(1) gauge groups is not directly linked to the Calabi-Yau manifold. It is instead connected through the "tangent bundle" of the Calabi-Yau manifold. The "tangent bundle" is an additional manifold created by collecting all the tangent planes on the manifold as shown in a 2-D example in Figure 24. The term "bundles" is used by mathematicians to express the "gauge fields" in physics. It has been shown that the tangent bundles (actually any bundles for the Cababi-Yau manifold) is equivalent to the Yang-Mills equations (with N = 1 supersymmetry) and thus makes the connection to the massless version of the Standard model. Later on it was also shown that any bundles (of the Calabi-Yau manifold) can satisfy the anomaly cancellation requirement if its second Chern class equals the second Cheren class of the tangent bundle.
3. Fermion Mass - Beyond getting the right particles, the Superstring theory has the potential to yield the mass of fermions and places itself above the Standard model. The mass of ferimon is proportional to the Yukawa coupling constant g, which is in turn determined by the triple product in the six dimensional Calabi-Yau space, where and are the fermion and Higgs fields respectively. Since the value of the fields depends on the location in the Calabi-Yau manifold, it is necessary to perform numerical integration over the six dimensional Calabi-Yau space to obtain the average by a process called "discretization" - a process that defies today's computer power. Another way to compute g is through "embedding" the Calabi-Yau manifold in a higher dimensional background space. But so far no one has been able to work out the coupling constant g or mass for any fermion. Anyway, this is one example of the attempts to derive fundamental constants in the 3+1 large dimensions from the 6 dimensional compactified space.

End of the Universe

According to the Superstring theory and by implication the Calabi-Yau Manifold there are at least two scenarios to hasten the end of this universe as we know it.


1. Metastable State - As shown in Figure 21, our universe is one of those sitting on a valley in the landscape. Most probably it would not be in the one with the lowest vacuum energy. There are two ways the state of our universe can decay into a lower one (Figure 25) :

Quantum Tunneling - Quantum theory allows penetration of a barrier from one state to another with a probability inversely proportional to the width of the barrier. The initial state will decay into the other state if it is at lower energy level. Thermal Activation - Another way to overcome a barrier is via thermal fluctuations, which will occur if the thermal energy is higher than the height of the potential barrier.

Figure 25 End of the Universe [view large image]

What would happen during and after the transition depends on the path and the state of the next incarnation. For example, it could have a negative cosmological constant, which would lead the universe into a "Big Crunch" - the reverse of the "Big Bang".

The entity sitting inside the valley in Figures 21 and 25 is referred to as "vacuum" because it is the product of the field equation in General Relativity - the deSitter universe, which is empty (except for a cosmological constant term) without any matter (visible or dark).


2. Decompactification - It has been mentioned that the compactified dimensions are held together by fluxes and branes. Such state is not stable since it needs energy to do the wrapping. Fluctuations provide a chance for the "explosion" in dimensions to happen; and when it does, everything from the tiniest particles to galactic superclusters, would instantly explode into the six expanding dimensions. At the end of the conflagration, spacetime would still be there (with ten large dimension now) but the laws of physics would change to an un-recognizable forms, nothing as we know it will survive. However, it is estimated that the chance for such kind of catastrophe to happen is about 1 to e^(10120).

Testing the Superstring Theory

	Recovering the lower effective theory is only one of the requirements for a successful new theory. It has to offer doable and falsifiable tests and to predict new phenomena in order to be acceptable as a better replacement. This website has listed some of the proposed tests many years ago. The followings
Figure 26 Tests [view large image]	summarize the five tests in "The Shape of Inner Space" (Figure 26), actually they have been mentioned already elsewhere in this website.

1. CBMR Anisotropy - CMBR (Cosmic Microwave Background Radiation) is the ancient and feeble microwave radiation emanating uniformly (supposedly) from all directions in the sky. Recently there is claim about anisotropy in the form of a cold and empty spot on the sky map from WMAP. It is argued that such "axis of evil" could be the result of interaction with another universe (vacuum) predicted by the Superstring theory (see more details in "WMAP Oddities").



2. Cosmic String - Cosmic strings are thought to be long, tube-like objects of high-energy material left over from the Big Bang. They are the most interesting type of topological defects because some cosmologists have suggested such material as an alternative source of the density irregularities, visible in CMBR. It is also an object of much speculation of time travel via the formation of closed time-like curve. Only later on it is linked to the Superstring theory. It is purported that the tiny string has been blown up to the enormous size as the result of cosmic expansion. However, WMAP measurements have shown that the actual form of the irregularities is inconsistent with those predicted by the string-based theories (see more controversial claims about "Cosmic String").



3. Calabi-Yau Warped Throat - It is suggested that the scenario of Big Bang and subsequent inflation can be replaced by reflection of a 3-D brane (representing our universe) at the tip of a Calabi-Yau warped throat. More details of such speculation and other stringy cosmologies can be found in "Pre-Big Bang Universes".



4. WIMP - WIMP stands for weakly interacting massive particles, which is a brainchild of the supersymmetry theory. It is a hot subject for current research in both theory and observation about dark matter. There were many controversial claims of detection but none conclusive. The discovery of such particles will indirectly support the Superstring theory via its association with supersymmetry (see more in "Nature of Dark Matter").



5. Gravity Leaking - In term of extra-dimensions, the feebleness of gravity (comparing to the other forces) is explained by leaking of its field into the other dimensions. It is suggested that the LHC (Large Hadron Collider) should be able to detection the leakage if there's any. The discovery will confirm the curial feature in Superstring theory - the extra-dimensions (see more in "Gravity in Extra-dimensions").

Apparent and Real Problems

There are many problems with Superstring theory as mentioned in a section about "Problems and Future Development" elsewhere within this website. Some of them are rather technical, the followings summarize a few of those conceptual ones.


• In spite of the many proposed tests as mentioned above, critic keeps maintaining that the Superstring theory is not testable. The reason could be the outlandish objects often associated with the tests. However, the anti-particle proposed by Dirac in 1928 is also bordering on fantasy before the positron had finally been detected in about 4 years. On the other hand the confirmation of the Superstring theory seems to take much longer (more than 20 years and counting), people just don't have the patience to wait that long (troubling either by psychological preconception or by monetary consideration).



• Another sore point about the Superstring theory is the richness of solutions. Early on in the investigation, researchers were looking for an unique universe with all the derived fundamental constants in agreement with observations. Their dreams were shattered by so many possible configurations. Actually it is common for a theory to have even an infinite number of solutions. For example, according to Newtonian mechanics there are infinite number of possible orbits for objects moving around the Sun, yet there are only 9 planets each circulating around in a specific way. The difficulty with Superstring theory is that we can observe only one universe (the one we live in) and would never know if other universes exist beyond the cosmic horizon (with different configurations).



• On the other hand, propoents often cite mathematical beauty in the Superstring theory as supporting evidence for its truth. They maintain that successful theories in the past always appear that way. However, beauty is a subjective feeling, which should never be relied upon by scientists. They also marvel at such wonder of mathematics, which seems to be so unreasonably well adopted to describe the physical world. Actually, it is the other way round. Physicists instead compress the languages of physics into mathematical formulas. In other word, not all the mathematical formulas have a correspondence to physical reality (only the selected few do).




The real problem with the Calabi-Yau manifold is related to the fact that it is based on a smooth and differentiable space (top of Figure 27). At the Planck scale of ~ 10-33 cm, spacetime becomes foamy similar to the picture at the bottom of Figure 27 (courtesy of the Uncertainty principle in quantum theory). Thus the classical geometry should be replaced by the non-commutative geometry in which the Calabi-Yau manifold has to be either modified or become obsolete altogether. That's the process of scientific method in which a better theory always supersedes the old one (see more in "Space-time Background Dependence").

Figure 27 Quantum Foam [view large image]

Meanwhile, effort is under way to salvage the beloved Calabi-Yau manifold by equating topological variation with quantum fluctuation. Hopefully the string in the Superstring theory will provide gentler disturbance to enable the scheme.

Nevertheless, the Calabi-Yau manifold is embraced all the way. It may not be a part of the ultimate solution in quantum gravity; but the manifold will help further development of the Superstring theory, and there is still much to be learnt as a pure geometric object. To emphasize the importance of geometry in mathematics and science, the inscription : "Let no one ignorant of geometry enter here" at the entrance of Plato's Academy is quoted at the end of the book (Figure 28).

Figure 28 Plato's Inscription [view large image]