1. Osmo Pekonen 1

2.  a unified ”theory of everything”  fundamental idea: elementary particles in the Planck scale (10 -35 m) appear as tiny vibrating ”strings” 2

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5.  gravitation: Newton’s Principia (1687)  Einstein’s general relativity (1915)  Einstein’s field equation is a statement about the geometry of 4-dimensional spacetime. 5

6. Unification of the three other fundamental forces:  electromagnetism (Maxwell’s equations ~1864)  weak force  strong force Fundamental concept of unification: symmetry group  electromagnetism  one foton  SU(1)  weak force  3 weak bosons W+, W−, Z  SU(2)  strong force  eight gluons  SU(3) Total number of dimensions of the unified symmetry group G = SU(1) × SU(2) × SU(3) is 1 + 3 + 8 = 12. 6

7.  C.N. Yang and Robert Mills: In a principal G-bundle minimize the Lagrangian to get the field equations 7

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10.  Bold idea: perhaps higher dimensions than 4 can give some additional leeway?  Pioneers of this idea:  Gunnar Nordström  Theodor Kaluza  Oskar Klein 10

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14. 14 …which has 6 × genus – 6 dimensions and is known in classical mathematics as the Riemann Moduli Space, resp. its covering space the Teichmüller Moduli Space

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18.  The number of necessary dimensions of spacetime is lowered to a slightly more palatable D=10 in Superstring Theory.  Why dont we see the extra 6 dimensions?  They are supposed to be wrapped up at each point of 4-dimensional spacetime as a mathematical structure called Calabi-Yau space (for specialists: compact Kählerian 6-manifold with vanishing Ricci curvature). 18

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20. …but there are only two relevant Lie groups of dimension 496: SO(32) and E 8 × E 8. Recall: the Standard Model only had 12 symmetries! The 248-dimensional E 8 is a most beautiful mathematical object. 20

21.  Are there 10 or 26 dimensions ?  Is the structure group SO(32) or E 8 × E 8 ?  There are myriads of different Calabi-Yau spaces. Which are the prefered ones?  Heterotic String Theory is a hybrid of Bosonic String Theory and Superstring Theory.  It was developed in 1985 by the ”Princeton string quartet” David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm. 21

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23. M-theory works in D=11 dimensions. At the expense of one extra dimension, it unifies the various 10-dimensional theories. There also exists D=12 dimensional F-theory due to Cumrun Vafa. Time also could be more than 1 dimensional. 23

24. Do we live in a holographic universe? In 1997 the Argentinian physicist Juan Maldacena discovered a new kind of duality AdS/CFT which works between a theory of gravity (AdS) and a lower dimensional gauge theory (CFT) without gravity. 24

25.  There is no experimental evidence whatsoever either for the existence of SUSY or the validity of any form of String Theory.  However, String Theory has already left a durable trace in many fields of mathematics (Witten is a Fields medalist).  String Theory could also turn out to be utterly wrong… or perhaps we might never know.  So there is some room for fantasy! 25

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