Cell Theory
1 Conifold is presented by the following.
n-dimensional complex projective space that has homogeneous coordinates (z1, z2, … , zn+1) is given the following condition.
|z1|2 +| z2| 2+ … + | zn+1|2 = r r>0
There emerges 2n+1-dimensional sphere S2n+1.
On arbitrary θ, when identification is done with the polar coordinate representation, Pn is presented.
(z1, z2, …, zn+1) ~ (eiθz1, eiθz2, … , eiθzn+1)
P1 that has line bundle’s direct sum O(-1) ⊕ O(-1) is conifold.
P1 has homogeneous coordinates (z1, z2) and line bundle coordinates (z3, z4).
Conifold that is also called local P1 is defined by the following.
|z1|2 +| z2| 2- | z3|2 - | z4|2 = r
When | z3|2 = | z4|2 = 0, |z1|2 +| z2| 2 = r is given as 2-dimensional sphere S2 that is called resolved conifold.
When complexification of parameter r becomes 0, there emerges conifold with singularity.
From here, deformed conifold is given by the blowing up resolved conifold.
Deformed conifold has 3-dimensional sphere S3 that wraps D6 brane.
Here S3 is identificated to word. Brane is identificated to grammar.
Further now topology R4 ×S3 is presented.
This topology is identificated to sentence.
[Reference]
Cell Theory / From Cell to Manifold / Tokyo June 2, 2007
Tokyo June 9, 2007
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