III.6 Calabi-Yau Manifolds: Conjecture, Manifolds in Physics, Eric Zaslow, The Calculus of Variations and III.7 Cardinality: Page 164.

The Calabi-Yau Manifold Calabi conjecture is that the metric notion amounts to a nonlinear partial differential equation in u, namely: det(gab’ + 2u/zab) = |f|2

Calabi proved the uniqueness and Yau proved the existence of the solution to this equation.  The metric definition of a Calabi-Yau manifold is uniquely determined by its Kähler structure and its complex orientation.

The space of metrics with holonomy group SU(n) on a manifold with complex orientation is in correspondence with the space of inequivalent Kähler structures.  The requirement of conformality and a vanishing ricci tensor are essentially equivalent. 

A Calabi_Yau manifold with its SU(n)-holonomy metric has vanishing Ricci tensor and is of interest in general relativity.  Calabi-Yau manifolds figure prominently in the leading theory of quantum gravity, string theory.  Quantum gravity is the incorporation of Einstein’s theory into the quantum theory of particles.

In string theory the fundamental objects are “strings”.   Worldsheets are used to describe the two dimensional trajectory of the strings.  String theory is is constructed from the two dimensional Riemann surfaces of worldsheets to a space-time manifold, M.  Although the number of possible metrics is infinite under the assumption of invariance under local changes of scale the dimensionality reduces to a finite number of conformally inequivalent metrics and the theory is well defined.  Adding the idea of supersymmetry means we have the space-time manifold M to be a complex manifold.

These two conditions together mean that M is a complex manifold with holonomy group SU(n) or a Calabi-Yau manifold.  The requirement of conformality and a vanishing Ricci tensor are essentially equivalent.  This means the theory matches Einsteins although without matter. 

Calabi-Yau manifolds are both symplectic and complex in the topological string theory.  This leads to two types of topological strings labeled A and B.  Mirror symmetry is the phenomenon that the A version of one Calabi-Yau manifold is related to the B version of another different Calabi-Yau manifold.  The B version is a mirror partner.  The mathematical consequences of this mirror symmetry are being investigated.

“The Calculus of Variations” is discussed under “Variational Methods”.

A finite set has cardinality equal to the number of elements in the set.  A countably infinite set is said to have cardinality  \aleph_0.  The reals are uncountable and have cardinality 2^{\aleph_0}=\aleph_1.   These numbers are called cardinals and they can be added, multiplied and raised to powers.  This will be discussed further under “set theory”.

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