The Quantum Structure of Space and Time (293 pages)

130 The Quantum Structure of Space and Time

where the LI parameterize the orientifold charge. In addition there are some more
constraints from K-theory. Chiral matter in bifundamental representations originate

from open strings located at the intersection of two stacks of D6-branes with a

multiplicity (generation) number given by the intersection number
3
Iab = x(xfY: - x,’Y,’). (6)
I=O
Counting all possible solutions of the D-brane equations (4) and (5) leads to a total
of 1.66. 10’ supersymmetric (4-stack) D-brane models on the 22 x 22 orientifold.
With this large sample of models we can ask the question which fraction of models
satisfy several phenomenological constraints that gradually approach the spectrum
of the supersymmetric MSSM. This is summarized in the following table: Therefore

Restriction

gauge factor U(3)

gauge factor U(2)/Sp(2)

No symmetric representations

Massless U( 1)y

Three generations of quarks

Three generations of leptons

Total

Factor

0.0816

0.992

0.839

0.423

2.92 x 10-5

1.62 x 10-3

1.3 x 10-9

only one in a billion models give rise to an MSSM like D-brane vacuum. Similar

results can be obtained for models with SU(5) GUT gauge group [as].

Finally we would like to pose the following question: is it possible to obtain

an entropy resp. a probability wave function for D-brane vacua? To answer this

question one might try to replace the D7-branes in IIB (D6-branes in IIA) by D5-

branes (D4-branes). This will lead to cosmic strings in D=4. So reformulating this

question would mean, can one associate an entropy to this type of cosmic string

solutions? In this way one could derive, besides the statistical counting factor, a

stringy probability measure for deriving the Standard Model from D-brane models.

Bi bliogrup h y

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Phys. Rev. D 68 (2003) 046005 [arXiv:hep-th/0301240].

[3] S. Gukov, C. Vafa and E. Witten, “CFT’s from Calabi-Yau four-folds,” Nucl. Phys.

B 584 (2000) 69 [Erratum-ibid. B 608 (2001) 4771 [arXiv:hep-th/9906070].

[4] T. R. Taylor and C. Vafa, “RR flux on Calabi-Yau and partial supersymmetry break-

ing,” Phys. Lett. B 474 (2000) 130 [arXiv:hep-th/9912152].

[5] S. B. Giddings, S. Kachru and J. Polchinski, “Hierarchies from fluxes in string com-

pactifications,” Phys. Rev. D 66 (2002) 106006 [arXiv:hep-th/0105097].